arXiv 1901.016092019-01nlin.SI

分离变量中的不变量：Yang-Baxter，缠绕和转移映射

Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps

Pavlos Kassotakis

发表信息：: SIGMA 15 (2019), 048, 36 pages
DOI：: 10.3842/SIGMA.2019.048

摘要：

我们提出了明确的形式，家庭的刘维可积映射在3个变量，所谓的三元组家庭的地图，我们提出了一个多领域的推广后者。通过对这类映射的不变量施加变量的可分性，我们得到了一般奇点位置的$H_{\rm I}$、$H_{\rm II}$和$H_{\rm III}^A$ Yang-Baxter映射.给出了两种不同的方法来获得纠缠杨-巴克斯特映射。第一种方法的结果是与$H_{\rm I}$、$H_{\rm II}$和$H_{\rm III}^A$ Yang-Baxter映射相关联的缠绕映射，而第二种方法得到了与整个$F$和$H$ -四元Yang-Baxter映射列表相关联的非周期缠绕映射.最后，我们证明了与杨-巴克斯特映射的$H$ -列表相关联的转移映射如何可以被认为是某些形式更简单的映射的$(k-1)$ -迭代.我们把这些地图作为扩展的转移地图，反过来，他们导致$k$ -点交替递归，可以被认为是交替版本的一些层次的离散潘勒韦方程。

英文摘要：

We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called triad family of maps and we propose a multi-field generalisation of the latter. We show that by imposing separability of variables to the invariants of this family of maps, the $H_{\rm I}$, $H_{\rm II}$ and $H_{\rm III}^A$ Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the $H_{\rm I}$, $H_{\rm II}$ and $H_{\rm III}^A$ Yang-Baxter maps, whereas by the second method we obtain non-periodic entwining maps associated with the whole $F$ and $H$-list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the $H$-list of Yang-Baxter maps can be considered as the $(k-1)$-iteration of some maps of simpler form. We refer to these maps as extended transfer maps and in turn they lead to $k$-point alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painlevé equations.

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